Consider the simulated RESCALE data file, which contains 100 observations of a response variable, Y, and a predictor variable X. By working through the following questions, you will find that a quadratic model fit to the data produces large variance inflation factors (VIFs) for X and X², whereas rescaling X first results in a model with considerably reduced VIFs. Nevertheless, the results from both models are essentially the same.

DATA LINK: https://file.io/nKlszCubgGax

Create a squared predictor term, X², and fit the quadratic model. Make a note of the t-statistic for b₂, the values of R^2, and the regression standard error, s, for the model. What is the value of s?

Calculate the VIFs for X and X² in the model from part (a). How much is it?

Create a rescaled predictor variable, Z = (X m_X)/s_X, where m_X and s_X are the sample mean and standard deviation of X, respectively. You can check that you have calculated Z correctly by checking that m_Z = 0 and s_Z = 1. Then create a squared rescaled predictor term, Z², and fit the quadratic model using the Zs. Make a note of the t-statistic for b₂, the values of R², and the regression standard error, s, for the model. What do you notice?